tMax=60;
n=20;
theta=Table[Unique[],{n}];
initialAngle=Pi/8;
r=Table[
First[x/.Solve[
tMax/(4 Sqrt[x]EllipticK[Sin[initialAngle/2]^2])==(20+i),x]],{i,1,n}];
equations=Flatten[Table[{
r[[i]]Sin[theta[[i]][t]]+r[[i]]^2theta[[i]]''[t]==0,
theta[[i]][0]==Pi/8,
theta[[i]]'[0]==0},
{i,1,n}]];
solution=NDSolve[equations,theta,{t,0,1.1tMax}];
Animate[
points=Table[First[
{r[[i]]Sin[theta[[i]][t]],-r[[i]]Cos[theta[[i]][t]]}/.solution/.t->T],{i,1,n}];
Graphics[{
Black, Sphere[#,0.005],
Gray,Line[{{0,0},#}]}&
/@points,
PlotRange->{{-0.8Max[r],0.8Max[r]},{0,-1.1Max[r]}}],
{T,0,1.1tMax},DefaultDuration->1.1tMax,
SaveDefinitions->True]